61 research outputs found
An analysis of a class of variational multiscale methods based on subspace decomposition
Numerical homogenization tries to approximate the solutions of elliptic
partial differential equations with strongly oscillating coefficients by
functions from modified finite element spaces. We present in this paper a class
of such methods that are very closely related to the method of M{\aa}lqvist and
Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and
Peterseim, these methods do not make explicit or implicit use of a scale
separation. Their compared to that in the work of M{\aa}lqvist and Peterseim
strongly simplified analysis is based on a reformulation of their method in
terms of variational multiscale methods and on the theory of iterative methods,
more precisely, of additive Schwarz or subspace decomposition methods.Comment: published electronically in Mathematics of Computation on January 19,
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Fractal homogenization of multiscale interface problems
Inspired by continuum mechanical contact problems with geological fault
networks, we consider elliptic second order differential equations with jump
conditions on a sequence of multiscale networks of interfaces with a finite
number of non-separating scales. Our aim is to derive and analyze a description
of the asymptotic limit of infinitely many scales in order to quantify the
effect of resolving the network only up to some finite number of interfaces and
to consider all further effects as homogeneous. As classical homogenization
techniques are not suited for this kind of geometrical setting, we suggest a
new concept, called fractal homogenization, to derive and analyze an asymptotic
limit problem from a corresponding sequence of finite-scale interface problems.
We provide an intuitive characterization of the corresponding fractal solution
space in terms of generalized jumps and gradients together with continuous
embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the
solution of the asymptotic limit problem and exponential convergence of the
approximating finite-scale solutions. Computational experiments involving a
related numerical homogenization technique illustrate our theoretical findings
Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations
We present globally convergent nonsmooth Schur-Newton methods for the solution
of discrete vector-valued Cahn-Hilliard equations with logarithmic and
obstacle potentials. The method solves the nonlinear set-valued saddle-point
problems as arising from discretization by implicit Euler methods in time and
first order finite elements in space without regularization. Efficiency and
robustness of the convergence speed for vanishing temperature is illustrated
by numerical experiments
A posteriori error estimates for elliptic variational inequalities
We derive hierarchical a posteriori error estimates for elliptic variational inequalities. The evaluation amounts to the solution of corresponding scalar local subproblems. We derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples
Adaptive monotone multigrid methods for some non-smooth optimization problems
We consider the fast solution of non-smooth optimization problems as resulting for example from the approximation of elliptic free boundary problems of obstacle or Stefan type. Combining well-known concepts of successive subspace correction methods with convex analysis, we derive a new class of multigrid methods which are globally convergent and have logarithmic bounds of the asymptotic convergence rates. The theoretical considerations are illustrated by numerical experiments
A Variational Approach to Particles in Lipid Membranes
A variety of models for the membrane-mediated interaction of particles in
lipid membranes, mostly well-established in theoretical physics, is reviewed
from a mathematical perspective. We provide mathematically consistent
formulations in a variational framework, relate apparently different modelling
approaches in terms of successive approximation, and investigate existence and
uniqueness. Numerical computations illustrate that the new variational
formulations are directly accessible to effective numerical methods
Numerical Homogenization of Fractal Interface Problems
We consider the numerical homogenization of a class of fractal elliptic
interface problems inspired by related mechanical contact problems from the
geosciences. A particular feature is that the solution space depends on the
actual fractal geometry. Our main results concern the construction of
projection operators with suitable stability and approximation properties. The
existence of such projections then allows for the application of existing
concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative
algebraic solvers with scale-independent convergence behavior for this class of
problems
Fractal homogenization of a multiscale interface problem
Inspired from geological problems, we introduce a new geometrical
setting for homogenization of a well known and well studied problem of an
elliptic second order differential operator with jump condition on a
multiscale network of interfaces. The geometrical setting is fractal and
hence neither periodic nor stochastic methods can be applied to the study of
such kind of multiscale interface problem. Instead, we use the fractal nature
of the geometric structure to introduce smoothed problems and apply methods
from a posteriori theory to derive an estimate for the order of convergence.
Computational experiments utilizing an iterative homogenization approach
illustrate that the theoretically derived order of convergenceof the
approximate problems is close to optimal
Fractal homogenization of a multiscale interface problem
Inspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergence of the approximate problems is close to optimal
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